Year 7 Angles – Exercise 1
1. Find the angles marked with letters.
2.
[JMC 2000 Q3] What is the value of
𝑥?
7.
[IMC 2011 Q9] In the diagram, 𝑋𝑌 is a
straight line. What is the value of 𝑥?
3.
[JMC 2015 Q6] What is the value of 𝑥
in this triangle?
8.
[JMC 2010 Q8] In a triangle with
angles 𝑥°, 𝑦°, 𝑧° the mean of 𝑦 and 𝑧 is
𝑥. What is the value of 𝑥?
4.
[JMC 2011 Q6] What is the sum of
the marked angles in the diagram?
9.
[JMC 1999 Q13] In the diagram,
∠𝑅𝑃𝑀 = 20° and ∠𝑄𝑀𝑃 = 70°.
What is ∠𝑃𝑅𝑆?
5.
6.
[JMC 1997 Q17] How big is angle 𝑥?
[IMC 2014 Q3] An equilateral triangle
is placed inside a larger equilateral
triangle so that the diagram has three
lines of symmetry. What is the value
of 𝑥?
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10. [JMC 2001 Q18] Triangle 𝑃𝑄𝑅 is
equilateral. Angle 𝑆𝑃𝑅 = 40°, angle
𝑇𝑄𝑅 = 35°. What is the size of the
marked angle 𝑆𝑋𝑇?
11. [JMC 2007 Q16] What is the sum of
the six marked angles?
12. [JMC 2015 Q16] The diagram shows a
square inside an equilateral triangle.
What is the value of 𝑥 + 𝑦?
13. [JMO 2001 A3] What is the value of 𝑥
in the diagram alongside?
14. [Kangaroo Pink 2010 Q9] In the
diagram, angle 𝑃𝑄𝑅 is 20°, and the
reflex angle at 𝑃 is 330°.The line
segments 𝑄𝑇 and 𝑆𝑈 are
perpendicular. What is the size of
angle 𝑅𝑆𝑃?
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15. [Kangaroo Pink 2006 Q8] The circle
shown in the diagram is divided into
four arcs of length 2, 5, 6 and 𝑥 units.
The sector with arc length 2 has an
angle of 30° at the centre. Determine
the value of 𝑥.
(Hint: An arc is a part of the line that makes up
the circle, i.e. part of the circumference. The
arc length grows in proportion to the angle at
the centre)
16. [SMC 2010 Q3] The diagram shows
an equilateral triangle touching two
straight lines. What is the sum of the
four marked angles?
Exercise 2 – Z/F/C angles and quadrilaterals
1. For each diagram (i) Find the missing angle(s) and (ii) List reasons for each answer.
a)
b)
2. Find the indicated unknown angles.
3. For rhombuses are evenly spaced around a point. Given the angle shown, find 𝑥.
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4. Find the angles indicated.
10. [IMC 2005 Q7] In the diagram, what
is the sum of the marked angles?
5. Find the angle 𝛼.
11. [JMO 2003 A9] Find the value of 𝑎 +
𝑏 + 𝑐.
6. The line 𝑃𝐵 bisects the angle 𝐴𝐵𝐶
(this means it cuts it exactly in half).
Determine the angle 𝐵𝑃𝐶.
7.
[JMC 2006 Q7] What is the value of
𝑥?
8.
[JMC 2007 Q9] In the diagram on the
right, 𝑆𝑇 is parallel to 𝑈𝑉. What is the
value of 𝑥?
9.
[JMC 2009 Q19] The diagram on the
right shows a rhombus 𝐹𝐺𝐻𝐼 and an
isosceles triangle 𝐹𝐺𝐽 in which 𝐺𝐹 =
𝐺𝐽. Angle 𝐹𝐽𝐼 = 111°. What is the
size of angle 𝐽𝐹𝐼?
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12. [SMC 2006 Q3] The diagram shows
overlapping squares. What is the
value of 𝑥 + 𝑦?
13. [Cayley 2012 Q2] In the diagram, 𝑃𝑄
and 𝑇𝑆 are parallel.
Prove that 𝑎 + 𝑏 + 𝑐 = 360.
14. [Kangaroo Grey 2005 Q20] Five
straight lines intersect at a common
point and five triangles are
constructed as shown. What is the
total of the 10 angles marked in the
diagram?
A 300°
B 450°
C
360°
D 600°
E
720°
Exercise 3 – Isosceles Triangles
1. Find the value of each variable.
2. [JMC 2014 Q10] An equilateral triangle is surrounded by three squares, as shown.
What is the value of 𝑥?
3. [JMC 2005 Q13] The diagram shows two equal squares. What is the
value of 𝑥?
4. [IMC 2010 Q6] In triangle 𝑃𝑄𝑅, 𝑆 is a point on 𝑄𝑅 such that
𝑄𝑆 = 𝑆𝑃 = 𝑃𝑅 and ∠𝑄𝑃𝑆 = 20°. What is the size of ∠𝑃𝑅𝑆?
5. [IMC 1999 Q8] In the diagram 𝑃𝑄 = 𝑃𝑅 = 𝑅𝑆.
What is the size of angle 𝑥?
6. [JMC 2008 Q19] In the diagram on the right, 𝑃𝑇 = 𝑄𝑇 = 𝑇𝑆, 𝑄𝑆 = 𝑆𝑅,
∠𝑃𝑄𝑇 = 20°. What is the value of 𝑥?
7. [Kangaroo Grey 2010 Q9] The diagram shows a quadrilateral 𝐴𝐵𝐶𝐷, in
which 𝐴𝐷 = 𝐵𝐶, ∠𝐶𝐴𝐷 = 50°, ∠𝐴𝐶𝐷 = 65° and ∠𝐴𝐶𝐵 = 70°. What is
the size of ∠𝐴𝐵𝐶?
8. [Cayley 2004 Q1] The “star” octagon shown in the diagram is beautifully
symmetrical and the centre of the star is at the centre of the circle. If angle
𝑁𝐴𝐸 = 110°, how big is the angle 𝐷𝑁𝐴?
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9. [IMC 2003 Q8] Lines 𝐴𝐵 and 𝐶𝐷 are parallel and 𝐵𝐶 = 𝐵𝐷. Given that 𝑥 is
an acute angle not equal to 60°, how many other angles in this diagram are
equal to 𝑥?
10. [IMC 1997 Q11] In the quadrilateral 𝐴𝐵𝐶𝐷, ∠𝐴𝐵𝐶 = 90°, ∠𝐵𝐴𝐷 = 70°, and
𝐴𝐵 = 𝐵𝐷 = 𝐵𝐶. What is the size of ∠𝐵𝐷𝐶?
11. [Kangaroo Pink 2013 Q9] The diagram shows an equilateral triangle 𝑅𝑆𝑇
and also the triangle 𝑇𝑈𝑉 obtained by rotating triangle 𝑅𝑆𝑇 about the
point 𝑇. Angle 𝑅𝑇𝑉 = 70°. What is angle 𝑅𝑆𝑉?
12. [IMC 2000 Q8] In the triangle 𝐴𝐵𝐶, 𝐴𝐷 = 𝐵𝐷 = 𝐶𝐷. What is the size of angle
𝐵𝐴𝐶?
13. In the diagram 𝐴𝐵 is parallel to 𝐶𝐷 and 𝐴𝐸 = 𝐴𝐶.
Determine ∠𝐶𝐴𝐸.
14. [TMC Regional 2008 Q9] If 𝐴𝐵 = 𝐵𝐶 = 𝐶𝐷 =
𝐷𝐸 = 𝐸𝐹 and angle 𝐴𝐸𝐹 = 75°, what is the size of
angle 𝐸𝐴𝐹?
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Exercise 4 – Algebraic Angles
1. Given the value of ∠𝐷𝐵𝐴 indicated, work out the value of ∠𝐷𝐵𝐶.
e.g. ∠𝐷𝐵𝐴 = 𝑦
 ∠𝐷𝐵𝐶 = 180 − 𝑦
a. ∠𝐷𝐵𝐴 = 𝑤
b. ∠𝐷𝐵𝐴 = 𝑥 + 10
c. ∠𝐷𝐵𝐴 = 2𝑥 − 30
d. ∠𝐷𝐵𝐴 = 90 + 2𝑥
e. ∠𝐷𝐵𝐴 = 130 + 𝑥 − 2𝑦
2. Given the values of ∠𝐴𝐵𝐶 and ∠𝐵𝐴𝐶 indicated, work out the value of ∠𝐴𝐶𝐵.
e.g. ∠𝐴𝐵𝐶 = 30°, ∠𝐵𝐴𝐶 = 40°  ∠𝐴𝐶𝐵 = 110°
a. ∠𝐴𝐵𝐶 = 𝑥, ∠𝐵𝐴𝐶 = 𝑥
b. ∠𝐴𝐵𝐶 = 𝑥, ∠𝐵𝐴𝐶 = 10°
c. ∠𝐴𝐵𝐶 = 𝑥 + 10°, ∠𝐵𝐴𝐶 = 150°
d. ∠𝐴𝐵𝐶 = 2𝑥 + 30°, ∠𝐵𝐴𝐶 = 20° − 5𝑥
e. ∠𝐴𝐵𝐶 = 4𝑥 − 20°, ∠𝐵𝐴𝐶 = 7𝑥 − 30°
3. Determine all the angles in each diagram in terms of 𝑥 and/or 𝑦.
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4. Write suitable expressions in terms of 𝑥 for each missing angle. Hence determine the
smallest angle in the diagram in each case.
a. Angle 𝐴𝐵𝐷 is 3 times bigger than angle 𝐷𝐵𝐶.
.
b. Angle 𝐵𝐶𝐴 is twice as large as angle 𝐵𝐴𝐶.
c. The angles 𝐴𝐶𝐵, 𝐴𝐵𝐶 and 𝐵𝐴𝐶 are in the ratio 3: 4: 5.
5. [Based on JMO 2005 B4] In this figure 𝐴𝐷𝐶 is a straight line and 𝐴𝐵 = 𝐵𝐶 = 𝐶𝐷. Also, 𝐷𝐴 =
𝐷𝐵. We wish to find the size of ∠𝐵𝐴𝐶. Let ∠𝐷𝐴𝐵 = 𝑥.
a) By using the information provided, determine all the remaining angles in the diagram in
terms of 𝑥.
b) By considering the three angles in Δ𝐷𝐶𝐵, hence determine ∠𝐵𝐴𝐶.
6. [Based on JMC 2011 Q23] The points 𝑆, 𝑇, 𝑈 lie on the sides of the triangle
𝑃𝑄𝑅, as shown, so that 𝑄𝑆 = 𝑄𝑈 and 𝑅𝑆 = 𝑅𝑇. ∠𝑇𝑆𝑈 = 40°.
a) Let ∠𝑇𝑆𝑅 = 𝑥° and ∠𝑈𝑆𝑄 = 𝑦°. What is the value of 𝑥 + 𝑦?
b) Using the information provided, find expressions for ∠𝑇𝑅𝑆 and ∠𝑈𝑄𝑆.
c) Hence find an expression for ∠𝑅𝑃𝑄.
d) Using your answer to part (a), hence find the value of ∠𝑅𝑃𝑄.
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7. [Based on TMC Final 2014 Q4] The triangle 𝐴𝐵𝐶 is isosceles with 𝐴𝐶 = 𝐵𝐶 as shown.
The point 𝐷 lies on the line 𝐵𝐶 such that the triangle 𝐴𝐵𝐷 is isosceles with 𝐴𝐵 = 𝐵𝐷
as shown. ∠𝐵𝐴𝐶 = 2 × ∠𝐵𝐶𝐴 and we wish to work out the value of ∠𝐴𝐷𝐶.
Suppose we let ∠𝐵𝐶𝐴 = 𝑥. Hence determine:
a) ∠𝐵𝐴𝐶
b) ∠𝐶𝐵𝐴
c) Hence by considering the triangle 𝐶𝐴𝐷, determine 𝑥.
d) Use the remaining information to determine ∠𝐴𝐷𝐶.
8. [JMO 2004 B1] In the rectangle ABCD, M and N are the midpoints of AB and CD respectively;
AB has length 2 and AD has length 1.
Given that ∠𝐴𝐵𝐷 = 𝑥°, calculate ∠𝐷𝑍𝑁 in terms of 𝑥.
9. [Based on JMO 2006 B3] In this diagram, Y lies on the line AC, triangles
ABC and AXY are right angled and in triangle ABX, AX = BX. The line
segment AX bisects angle BAC and angle AXY is seven times the size of
angle XBC.
Let ∠𝑋𝐵𝐶 = 𝑥. In terms of 𝑥, find expressions for:
a) ∠𝐴𝑋𝑌
b) ∠𝐵𝑋𝑌 (hint: the lines 𝑋𝑌 and 𝐵𝐶 are parallel)
c) ∠𝑋𝐴𝑌
d) ∠𝐵𝐴𝑋
e) ∠𝐵𝑋𝐴 (using the fact that Δ𝐵𝑋𝐴 is isosceles)
f) Optional: By considering angles around a suitable point, hence find ∠𝐴𝐵𝐶.
10. [JMC 2003 Q23] In the diagram alongside, 𝐴𝐵 = 𝐴𝐶 and 𝐴𝐷 = 𝐶𝐷.
How many of the following statements are true for the angles marked?
𝑤=𝑥
𝑥 + 𝑦 + 𝑧 = 180
𝑧 = 2𝑥
A all of them
D none of them
B two
C one
E it depends on 𝑥
11. [JMC 2005 Q23] In the diagram, triangle 𝑋𝑌𝑍 is isosceles, with 𝑋𝑌 =
𝑋𝑍. What is the value of 𝑟 in terms of 𝑝 and 𝑞?
A
1
(𝑝
2
− 𝑞)
D 𝑝+𝑞
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B
1
(𝑝
2
+ 𝑞)
C 𝑝−𝑞
E Impossible to determine
12. [JMC 2015 Q25] The four straight lines in the diagram are such that
𝑉𝑈 = 𝑉𝑊. The sizes of ∠𝑈𝑋𝑍, ∠𝑉𝑌𝑍 and ∠𝑉𝑍𝑋 are 𝑥°, 𝑦° and 𝑧°.
Which of the following equations gives 𝑥 in terms of 𝑦 and 𝑧?
A 𝑥 =𝑦−𝑧
B 𝑥 = 180 − 𝑦 − 𝑧
D 𝑥 = 𝑦 + 𝑧 − 90
E 𝑥=
𝑦−𝑧
2
𝑧
C 𝑥 =𝑦−2
13. [JMO 2008 B3] In the diagram ABC and APQR are congruent triangles. The side PQ passes
through the point D and ∠𝑃𝐷𝐴 = 𝑥°. Find an expression for ∠𝐷𝑅𝑄 in terms of x.
14. [JMO 2008 A9] In the diagram, 𝐶𝐷 is the bisector of angle 𝐴𝐶𝐵.
Also 𝐵𝐶 = 𝐶𝐷 and 𝐴𝐵 = 𝐴𝐶. What is the size of angle 𝐶𝐷𝐴?
15. [JMO 2010 A10] Inn the diagram, 𝐽𝐾 and 𝑀𝐿 are parallel. 𝐽𝐾 = 𝐾𝑂 =
𝑂𝐽 = 𝑂𝑀 and 𝐿𝑀 = 𝐿𝑂 = 𝐿𝐾. Find the size of angle 𝐽𝑀𝑂.
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Exercise 5a – Simple Proofs
c. Points 𝐴, 𝐵 and 𝐶 lie in that
order on a straight line.
A point 𝐷, not on the line, is
placed such that
𝐴𝐷 = 𝐴𝐵, 𝐵𝐷 = 𝐵𝐶, and
∠𝐵𝐷𝐶 = 25°.
d. Points 𝑇 and 𝑈 lie outside a
parallelogram 𝑃𝑄𝑅𝑆, and are
such that triangles 𝑅𝑄𝑇 and
𝑆𝑅𝑈 are equilateral and lie
wholly outside the
parallelogram.
Note that where relevant, full proofs are
required – a sequential list of justified
statements, not just a diagram with angles put
in.
1. a) Prove ∠𝑅𝑄𝐶 = 55°. Your proof
should only require one line and be in
the form “∠𝑎𝑛𝑔𝑙𝑒 =
𝑣𝑎𝑙𝑢𝑒 (𝑟𝑒𝑎𝑠𝑜𝑛)”
3.
Prove the following:
a. ∠𝐵𝐶𝐴 = 20°
b) Prove that ∠𝐸𝐹𝐻 = 85°. (Your
proof should consist of two lines)
b.
∠𝐷𝐸𝐹 = 35°
c) Prove that ∠𝐵𝐶𝐷 = 2𝑥. (Your proof
should consist of three lines)
Exercise 5b – More Complex Proofs
2. Draw diagrams which satisfy the
following criteria, ensuring you note
where lines are of equal length or
parallel. You do NOT need to find any
angles.
a. 𝐴𝐵𝐶 is a right-angled triangle
such that ∠𝐶𝐴𝐵 = 90°. 𝐷 is a
point on 𝐵𝐶 such that 𝐴𝐷 =
𝐵𝐷.
b. 𝐴𝐵𝐶 is an isosceles triangle
where 𝐴𝐵 = 𝐴𝐶. 𝐷 is a point
that lies inside the triangle
such that 𝐵𝐶𝐷 is equilateral.
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1. Prove the following.
a. 𝐴𝐷𝐶 and 𝐴𝐵𝐶 are rightangled triangles where
∠𝐴𝐶𝐷 = 10° and ∠𝐷𝐶𝐵 =
70°. Prove that triangle 𝐴𝐵𝐶
is isosceles.
b. 𝐴𝐶 = 𝐵𝐶, 𝐴𝐶𝐷 is a straight
line, ∠𝐵𝐶𝐴 = 40° and
∠𝐵𝐶𝐸 = 70° as per the
diagram. Prove that 𝐴𝐵 and
𝐶𝐸 are parallel.
2. As before, 𝐴𝐶 = 𝐵𝐶. Suppose also
that 𝐶𝐸 bisects the angle 𝐵𝐶𝐷, but
unlike before, no angles are known.
By introducing a suitable variable (you
may wish to start your proof “Let
∠𝑎𝑛𝑔𝑙𝑒 = 𝑥”) prove that 𝐴𝐵 and 𝐶𝐸
are parallel.
3. [JMO 2001 B3] In the diagram, B is the
midpoint of AC and the lines AP, BQ
and CR are parallel. The bisector of
∠𝑃𝐴𝐵 meets BQ at Z.
Draw a diagram to show this, and join
Z to C.
(i) Given that ∠𝑃𝐴𝑍 = 𝑥°, find
∠𝑍𝐵𝐶 in terms of x.
(ii) Show that CZ bisects ∠𝐵𝐶𝑅.
4. [JMO 2015 B2] The diagram shows
triangle 𝐴𝐵𝐶, in which ∠𝐴𝐵𝐶 = 72°
and ∠𝐶𝐴𝐵 = 84°. The point 𝐸 lies on
𝐴𝐵 so that 𝐸𝐶 bisects ∠𝐵𝐶𝐴. The
point 𝐹 lies on 𝐶𝐴 extended. The
point 𝐷 lies on 𝐶𝐵 extended so that
𝐷𝐴 bisects ∠𝐵𝐴𝐹.
Prove that 𝐴𝐷 = 𝐶𝐸.
5. [JMO 2011 B4] In a triangle ABC, M
lies on AC and N lies on AB so that
∠𝐵𝑁𝐶 = 4𝑥°, ∠𝐵𝐶𝑁 = 6𝑥° and
∠𝐵𝑀𝐶 = ∠𝐶𝐵𝑀 = 5𝑥°. Prove that
triangle ABC is isosceles.
6. [JMO 2015 B4] The point 𝐹 lies inside
the regular pentagon 𝐴𝐵𝐶𝐷𝐸 so that
𝐴𝐵𝐹𝐸 is a rhombus. Prove that 𝐸𝐹𝐶
is a straight line. (Hint: the interior
angle of a pentagon is 108°)
7. [Hamilton 2006 Q2] In triangle ABC,
∠𝐴𝐵𝐶 is a right angle. Points P and Q
lie on AC; BP is perpendicular to AC;
BQ bisects ∠𝐴𝐵𝑃. Prove that CB = CQ.
8. 𝐴𝐵𝐶 is a triangle such that ∠𝐴𝐵𝐶 =
90° and 𝐷 is a point on the line 𝐴𝐶
such that 𝐵𝐷 = 𝐶𝐷. By letting
∠𝐷𝐶𝐵 = 𝑥 or otherwise, prove that
𝐶𝐷 = 𝐷𝐴.
(hint: prove that Δ𝐴𝐵𝐷 is isosceles first).
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Solutions – Exercise 1
1. 𝑎 = 64°, 𝑏 = 70°, 𝑐 = 60°,
𝑑 = 112°, 𝑒 = 35°
2. 28
3. 50
4. 360
5. 104
6. 150
7. 170
8. 60
9. 130
10. Solution:135
11. Solution:1440
12. 150
13. 15
14. 40
15. 11
16. 240
Exercise 2
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
–
–
120°
–
71°
55°
75°
86°
27
360°
210
270
Suppose we add a line horizontal with
𝑅 as shown, where 𝑏 is split into 𝑏1
and 𝑏2 .
𝑎 and 𝑏1 are cointerior thus 𝑎 + 𝑏1 =
180°.
𝑏2 and 𝑐 are cointerior thus 𝑏2 + 𝑐 =
180°
Thus
𝑎+𝑏+𝑐
= 𝑎 + 𝑏1 + 𝑏2 + 𝑐
= 180° + 180° = 360°
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Solutions – Exercise 3
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
𝑥 = 46°, 𝑦 = 74°, 𝑧 = 20°
30
140
40°
108°
35
55°
20°
4
65°
35°
90°
12°
21°
See slides for solutions to Exercises 4 and 5.
Angle Cheatsheet
Possible angles justifications:









“Angles on a straight line sum to 180°.”
“Angles in a triangle sum to 180°.”
“Angles around a point sum to 360°.”
“Base angles of an isosceles triangle are equal.
“Alternate angles are equal.”
“Corresponding angles are equal.”
“Vertically opposite angles are equal.”
“Cointerior angles sum to 180°.”
“Exterior angle of a triangle is the sum of the two other interior angles.”
Proof Format
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If you introduced any variables, start for example with: “Let ∠𝐴𝐵𝐶 = 𝑥.”
For statements about angles, it may be helpful to write in the form:
“∠𝑎𝑛𝑔𝑙𝑒 = 𝑣𝑎𝑙𝑢𝑒 (𝑟𝑒𝑎𝑠𝑜𝑛)”
where your reason is in the list above.
Where necessary have an appropriate conclusion:
Type of Proof
What you need to show
Some triangle is isosceles
Either show two angles are equal
or two sides are equal.
Show that the angles at 𝐵 sum to
180°, i.e. ∠𝐴𝐵𝐶 = 180°.
𝐴𝐵𝐶 is a straight line.
𝐴𝐵 = 𝐵𝐶
Two lines are parallel.
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We can show lengths are equal if
angles are equal, e.g. the triangle
is isosceles.
Show either the corresponding or
alternate angles on a connecting
line are equal.
How you might conclude your
proof
“∠𝐴𝐵𝐶 = ∠𝐵𝐴𝐶 therefore Δ𝐴𝐵𝐶
is isosceles.”
“∠𝐴𝐵𝐷 + ∠𝐷𝐵𝐶 = 180° therefore
𝐴𝐵𝐶 is a straight line.”
“∠𝐴𝐵𝐶 = ∠𝐴𝐶𝐵 and therefore
Δ𝐴𝐵𝐶 is isosceles. Thus 𝐴𝐵 = 𝐴𝐶.”
(With respect to diagram on left)
“∠𝐵𝐴𝐶 = ∠𝐷𝐶𝐸 and 𝐴𝐶𝐸 is a
straight line. Therefore 𝐴𝐵 and 𝐷𝐸
are parallel.”